Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition

Pal, Swadesh; Petrovskii, Sergei; Ghorai, S.; Banerjee, Malay

doi:10.1016/j.cnsns.2020.105478

Cited by 16 publications

(10 citation statements)

References 34 publications

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“…In the mathematical model, the diffusion term captures such types of movements phenomena, and it has been widely accepted in the spatio-temporal prey-predator model. Along with this, researchers have been studied different type of nonlocal interactions in the preypredator model [2,3,4,5,6]. Following [4], we consider the nonlocal interaction in the intraspecific competition of the prey population, and in this case, the corresponding integro-differential reaction-diffusion model is given by…”

Section: Mathematical Modelmentioning

confidence: 99%

“…The usual reaction-diffusion system can not capture such type of phenomena; however, a reaction-diffusion system with nonlocal interaction can. Different type of nonlocal models have been studied for prey-predator interactions in different circumstances, e.g., nonlocal dispersal [2], nonlocal consumption of resources [3], nonlocal intraspecific competition [4,5,6,7], etc. The use of reaction-diffusion equations with nonlocal interaction terms is a newly emerged area of research, whereas this modelling approach is not limited by the models of population interactions.…”

Section: Introductionmentioning

confidence: 99%

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Pattern alternations induced by nonlocal interactions

Pal¹,

Melnik²,

Banerjee³

2022

Preprint

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Pattern formation is a visual understanding of the dynamics of complex systems. Patterns arise in many ways, such as the segmentation of animals, bacterial colonies during growth, vegetation, chemical reactions, etc. In most cases, the long-range diffusion occurs, and the usual reaction-diffusion (RD) model can not capture such phenomena. The nonlocal RD model, on the other hand, can fill the gap. Analytical derivation of the amplitude equations (AE) for an RD system is a valuable tool to predict the pattern selections, in particular, the stationary Turing patterns when they occur. In this paper, we analyze the conditions for the Turing bifurcation for the nonlocal model and also derive the AE for the nonlocal RD model near the Turing bifurcation threshold to describe the reason behind the pattern selections. This derivation of the AE is not only limited to the nonlocal prey-predator model, as shown in our representative example but also can be applied to other nonlocal models near the Turing bifurcation threshold. The analytical prediction agrees with numerical simulation near the Turing bifurcation threshold. Moreover, the analytical and numerical results fit each other well even more remote from the Turing bifurcation threshold for the small values of the nonlocal parameter but not for the higher values.

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Section: Mathematical Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pattern alternations induced by nonlocal interactions

Pal¹,

Melnik²,

Banerjee³

2022

Preprint

Self Cite

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“…Along this way, there has been a great deal of accumulated achievements to enrich pattern formations within coupled reaction-diffusion systems. Such as, the stripe patterns, the stripe and the spotted mixed patterns in a one-dimensional region [5,6], and more complex hexagonal patterns, the labyrinthine-like stripe patterns, Eckhaus patterns, chaos in a two-dimensional region [7,8]. What is more, the spatiotemporal patterns also have been investigated where the Turing mode and the non-Turing mode intersect by reaction-diffusion equations.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal patterns in a general networked activator–substrate model

Chen

Zheng

et al. 2021

Nonlinear Dyn

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For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one

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“…As far as we know, although the existence of spatiotemporal phenomena has been studied to much extent through the Turing-Hopf bifurcation in many population dynamical models [1,4,7,21,27,32,33], there are few or no results in Turing-Hopf bifurcations for the competition system. Thus investigating what cause the occurrence of the Turing-Hopf bifurcation and how the spatiotemporal pattern can be formed is meaningful and worth exploring.…”

mentioning

confidence: 99%

“…By (27), (28) and the continuity of F (w), there must exist w > 0 such that F (w) = 0 holds. This completes the proof.…”

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confidence: 99%

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

Chen¹,

Jiang²

2021

DCDS-B

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<p style='text-indent:20px;'>We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.</p>

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Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition

Cited by 16 publications

References 34 publications

Pattern alternations induced by nonlocal interactions

Pattern alternations induced by nonlocal interactions

Spatiotemporal patterns in a general networked activator–substrate model

Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

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